Surds

Lesson

A binomial is an expression of the form $A+B$`A`+`B`, containing two terms. Changing the sign of the second term gives us the binomial $A-B$`A`−`B`, which we call a conjugate for the original binomial $A+B$`A`+`B`.

If we then try to find a conjugate for the binomial $A-B$`A`−`B` by changing the sign of the second term, we obtain the original binomial $A+B$`A`+`B`. That is, any binomial is a conjugate of its own conjugate. We often refer to two such binomials as a conjugate pair.

Notice that the product of a conjugate pair has a familiar form $\left(A+B\right)\left(A-B\right)$(`A`+`B`)(`A`−`B`) which is the factorised form of the difference of two squares $A^2-B^2$`A`2−`B`2. This observation motivates us to look at binomials containing surds - note that the expression $A^2-B^2$`A`2−`B`2 will be rational even if the terms $A$`A` or $B$`B` are square roots.

Consider a binomial such as $1+\sqrt{2}$1+√2. We can find a conjugate for this expression in the same way - by switching the sign of the second term. Doing so, we find that $1-\sqrt{2}$1−√2 is a conjugate for $1+\sqrt{2}$1+√2.

The process is the same even if the expression is more complicated, such as $\sqrt{x}-4\sqrt{3}$√`x`−4√3. A conjugate for this expression would be $\sqrt{x}+4\sqrt{3}$√`x`+4√3.

Summary

For any binomial expression $A+B$`A`+`B`, we can find a conjugate $A-B$`A`−`B` by changing the sign of the second term.

A binomial and its conjugate are sometimes called a conjugate pair.

A side note

We can rewrite the binomial $A+B$`A`+`B` in the equivalent form $B+A$`B`+`A` by changing the order of the terms. By doing so we can see that $B-A$`B`−`A` is also a conjugate for this expression, as well as $A-B$`A`−`B`.

That is, a binomial has two possible conjugates (since there are two orders in which the binomial can be written).

Determine a conjugate for $1+\sqrt{10}$1+√10.

Determine a conjugate for $\sqrt{5}-\sqrt{x}$√5−√`x`.

Manipulate complex numbers and present them graphically

Apply the algebra of complex numbers in solving problems